Adv. Geom. 4 (2004), 41–59
Advances in Geometry
( de Gruyter 2004
Linear systems with multiple base points in P 2
Brian Harbourne and Joaquim Roé
(Communicated by R. Miranda)
Abstract. Conjectures for the Hilbert function hðn; mÞ and minimal free resolution of the mth
symbolic power Iðn; mÞ of the ideal of n general points of P 2 are verified for a broad range of
values of m and n where both m and n can be large, including (in the case of the Hilbert function) for infinitely many m for each square n > 9 and (in the case of resolutions) for infinitely
many m for each even square n > 9. All previous results require either that n be small or be a
square of a special form, or that m be small compared to n. Our results are based on a new
approach for bounding the least degree among curves passing through n general points of P 2
with given minimum multiplicities at each point and for bounding the regularity of the linear
system of all such curves. For simplicity, we work over the complex numbers.
1 Introduction
Consider the ideal I ðn; mÞ H R ¼ C½P 2 generated by all forms having multiplicity
at least m at n given general points of P 2 . This is a graded ideal, and thus we
can consider the Hilbert function hðn; mÞ whose value at each nonnegative integer
t is the dimension hðn; mÞðtÞ ¼ dim I ðn; mÞt of the homogeneous component
Iðn; mÞ
t
mþ1
n
,
of I ðn; mÞ of degree t. It is well known that hðn; mÞðtÞ d max 0; tþ2
2
2
with equality for t su‰ciently large. Denote by aðn; mÞ the least degree
t
such
that
hðn; mÞðtÞ > 0 and by tðn; mÞ the least degree t such that hðn; mÞðtÞ ¼ tþ2
n mþ1
2
2 ;
we refer to tðn; mÞ as the regularity of I ðn; mÞ.
For n c 9, the Hilbert function [29] and minimal free resolution [17] of I ðn; mÞ are
known. For n > 9, there are in general only conjectures:
Conjecture 1.1. Let n d 10 and m d 0; then:
pffiffiffi
(a) aðn; mÞ d m n;
(b) hðn; mÞðtÞ ¼ max 0; tþ2
n mþ1
for each integer t d 0; and
2
2
(c) the minimal free resolution of I ðn; mÞ is an exact sequence
0 ! R½a 2 d l R½a 1 c ! R½a 1 b l R½a a ! I ðn; mÞ ! 0;
where a ¼ aðn; mÞ, a ¼ hðn; mÞðaÞ, b ¼ maxðhðn; mÞða þ 1Þ 3hðn; mÞðaÞ; 0Þ, c ¼